Question 1139652
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I am only vaguely familiar with this subject; so my response might not be mathematically rigorous.<br>
We are to show that every polynomial of degree 3 (element of P³) can be formed by a linear combination of {{{x^3+x^2}}}, {{{2x^2+1}}}, {{{x}}}, and {{{3}}}.<br>
Let an arbitrary third degree polynomial be {{{ax^3+bx^2+cx+d}}}.  Show that the polynomial can be expressed as {{{p(x^3+x^2)+q(2x^2+1)+r(x)+s(3)}}}.<br>
{{{p(x^3+x^2)+q(2x^2+1)+r(x)+s(3) = (p)x^3+(p+2q)x^2+(r)x+(q+3s)}}}<br>
We need to show that we can express the arbitrary coefficients a, b, c, and d in terms of p, q, r, and s.  It is easy to see that this is possible by equating the coefficients of each term in the two polynomials:<br>
{{{ax^3+bx^2+cx+d = (p)x^3+(p+2q)x^2+(r)x+(q+3s)}}}<br>
a = p; b = p+2q; c = r; d = q+3s<br>
Example: {{{7x^3-5x^2+4x-3}}}  (a = 7; b = -5; c = 4; d = -3)<br>
a = p = 7;
b = p+2q = 7+2q = -5  -->  q = -6;
c = r = 4;
d = q+3s = -6+3s = -3  -->  s = 1.<br>
{{{7(x^3+x^2)-6(2x^2+1)+4(x)+1(3) = 7x^3+(7-12)x^2+4x+(-6+3) = 7x^3-5x^2+4x-3}}}